Complexity in linear systems: a chaotic linear operator on the space of odd \(2 \pi\)-periodic functions (Q2012812)

Summary: Not just nonlinear systems but infinite-dimensional linear systems can exhibit complex behavior. It has long been known that twice the backward shift on the space of square-summable sequences \(l_2\) displays chaotic dynamics. Here we construct the corresponding operator \(\mathcal{C}\) on the space of \(2 \pi\)-periodic odd functions and provide its representation involving a Principal Value Integral. We explicitly calculate the eigenfunctions of this operator, as well as its periodic points. We also provide examples of chaotic and unbounded trajectories of \(\mathcal{C}\).